Question 8.4: Boundary Conditions Between a Pin and Hole Problem: An overh...
Boundary Conditions Between a Pin and Hole
Problem: An overhung beam is supported on two pins in bearings. Determine a suitable boundary condition scenario for its analysis and estimate its maximum deflection, maximum stress, and spring rate at the load point.
Given: The beam dimensions in Figure 8-19 (p. 499) are: a = 8, l = 20, b = 0.75, h = 2, and d = 0.5 in. The load F = 100 lb. The beam and pins are steel and the bearings are bronze on steel. The beam cross section is constant over its length except for the pin holes, which are all 0.5-in-dia.
Assumptions: Loading and supports reactions are coplanar. Pin supports are significantly stiffer than the beam.
1 First solve the beam by classical closed-form methods as shown in Example 4-6 (p.169). The equations for this overhung beam case can also be found in Appendix B, Figure B3, part (a). The maximum stress will be at the right support of Figure 8-19 where the moment magnitude is at its maximum: M = F(l – a) = 1200 lb-in. The bending stress at the outer fiber at x = a is:
\sigma=\frac{M c}{I}=\frac{1200(1)}{\frac{0.75\left(2^3-0.5^3\right)}{12}}=\frac{1200}{0.4922}=2438 psi (a)
Note that the I of the cross section was reduced by the effect of the hole at x = a.
2 The maximum deflection will be at the right end of the beam and is found from the equation in Appendix B, Figure B3, part (a) with x = 20, a = 8, l = 20, and I = 0.50 for the full cross section without any reduction for holes:
\begin{aligned} y &=\frac{F}{6 a E I}\left[(a-l) x^3-a\langle x-l\rangle^3+l\langle x-a\rangle^3+l\left(-l^2+3 a l-2 a^2\right) x\right] \\ &=-0.0064 \text { in } \end{aligned} (b)
3 The first trial constrains all the nodes around the circumference of each hole in both x and y directions. The mesh and boundary conditions around one hole of the beam and the undeflected and deflected mesh are shown in Figure 8-20. Note that there is no movement of any of the nodes on the hole circumference.
4 The second trial places a node at the hole center and connects it with rigid body elements to all the nodes on the hole circumference. These are sometimes called “kinematic couplings.” This technique effectively constrains the nodes on the circumference to have no motion in the radial direction but leaves them free to rotate about the hole center.* Figure 8-21a shows the mesh around one hole with these constraints applied. The nodes at both hole centers are fixed in x and y because the pins are on the beam’s neutral axis, which does not change length when the beam deflects. Figure 8-21b shows the undeflected and deflected meshes superposed. Note the rotation of the nodes on the hole circumference when the beam deflects, allowing the part to rotate on the “pin” as it deflects.
5 Figure 8-22 shows the deflected shapes and maximum deflections of the two beam models and their von Mises stress distributions. The deflection calculated by the “kinematic coupling” method was 0.0066 in, very close to that calculated by beam theory in step 2.
6 The deflection calculated by the FEA “circumferential fixation” method was 0.0049 in, a 23% error. Constraining the entire hole circumference makes the beam much stiffer.
7 The spring rate for the beam is easily calculated from the deflection and applied force as k = F / y = 100 / 0.0066 = 15 152 lb/in.
* If your FEA package does not provide rigid body elements, the same effect can be obtained by setting up a cylindrical coordinate system at the hole center and then constraining the radial coordinates of the nodes on the hole circumference while leaving their angles unconstrained.
